Codeforces Round #396 (Div. 2) E

Mahmoud and Ehab live in a country with n cities numbered from 1 to n and connected by n - 1 undirected roads. It's guaranteed that you can reach any city from any other using these roads. Each city has a number ai attached to it.

We define the distance from city x to city y as the xor of numbers attached to the cities on the path from x to y (including both x and y). In other words if values attached to the cities on the path from x to y form an array p of length l then the distance between them is $\text{[math]}$ , where $\text{[math]}$ is bitwise xor operation.

Mahmoud and Ehab want to choose two cities and make a journey from one to another. The index of the start city is always less than or equal to the index of the finish city (they may start and finish in the same city and in this case the distance equals the number attached to that city). They can't determine the two cities so they try every city as a start and every city with greater index as a finish. They want to know the total distance between all pairs of cities.

Input

The first line contains integer n (1 ≤ n ≤ 10⁵) — the number of cities in Mahmoud and Ehab's country.

Then the second line contains n integers a1, a2, ..., an (0 ≤ ai ≤ 10⁶) which represent the numbers attached to the cities. Integer ai is attached to the city i.

Each of the next n - 1 lines contains two integers u and v (1 ≤ u, v ≤ n, u ≠ v), denoting that there is an undirected road between cities uand v. It's guaranteed that you can reach any city from any other using these roads.

Output

Output one number denoting the total distance between all pairs of cities.

Examples

input

output

input

output

input

output

Note

A bitwise xor takes two bit integers of equal length and performs the logical xor operation on each pair of corresponding bits. The result in each position is 1 if only the first bit is 1 or only the second bit is 1, but will be 0 if both are 0 or both are 1. You can read more about bitwise xoroperation here: https://en.wikipedia.org/wiki/Bitwise_operation#XOR.

In the first sample the available paths are:

city 1 to itself with a distance of 1,
city 2 to itself with a distance of 2,
city 3 to itself with a distance of 3,
city 1 to city 2 with a distance of $\text{[math]}$ ,
city 1 to city 3 with a distance of $\text{[math]}$ ,
city 2 to city 3 with a distance of $\text{[math]}$ .

The total distance between all pairs of cities equals 1 + 2 + 3 + 3 + 0 + 1 = 10.

题意：求树每两个点的异或和（包括本身）

解法：

1 dp[i][0]表示经过i点，异或值为0的有多少,dp[i][1]表示经过i点，异或值为1的有多少

2 因为是异或，每一位都是独立的，我们就把它们按照每一位进行计算，然后根据乘法原理加起来

3 这个代码包括很多技巧

4 自己本身就加自己就好了

 1 #include<bits/stdc++.h>
 2 using namespace std;
 3 const int maxn=1e5+10;
 4 vector<long long>Ve[maxn];
 5 int n;
 6 long long dp[maxn][2];
 7 long long sum;
 8 int a[maxn];
 9 long long ans;
10 void dfs(int i,int pr,int bit){
11     int pos=(a[i]>>bit)&1;
12     dp[i][pos]=1;
13     dp[i][pos^1]=0;
14     long long sit=0;
15     for(int x=0;x<Ve[i].size();x++){
16         int v=Ve[i][x];
17         if(v==pr){
18             continue;
19         }
20         dfs(v,i,bit);
21         sit+=(long long)(dp[v][0]*dp[i][1]+dp[i][0]*dp[v][1]);
22         dp[i][pos^0]+=dp[v][0];
23         dp[i][pos^1]+=dp[v][1];
24     }
25     ans+=(sit<<bit);
26    // cout<<ans<<endl;
27 }
28 int main(){
29     cin>>n;
30     for(int i=1;i<=n;i++){
31         cin>>a[i];
32         sum+=a[i];
33     }
34     for(int i=1;i<n;i++){
35         int x,y;
36         cin>>x>>y;
37         Ve[x].push_back(y);
38         Ve[y].push_back(x);
39     }
40     for(int i=0;i<=26;i++){
41         dfs(1,0,i);
42     }
43     cout<<sum+ans<<endl;
44     return 0;
45 }